First numerical investigation of a conjecture by N. N. Nekhoroshev about stability in quasi-integrable systems

Abstract

We investigate numerically a conjecture by N. N. Nekhoroshev about the influence of a geometric property, called steepness, on the long term stability of quasi-integrable systems. In a Nekhoroshev's 1977 paper, it is conjectured that, among the steep systems with the same number ν of frequencies, the convex ones are the most stable, and it is suggested to investigate numerically the problem. Following this suggestion, we numerically study and compare the diffusion of the actions in quasi-integrable systems with different steepness properties in a large range of variation of the perturbation parameter ɛ and different dimensions of phase space corresponding to ν = 3 and ν = 4 (ν ≤ 2 is not significant for the conjecture). For six dimensional maps (ν = 4), our numerical experiments perfectly agree with the Nekhoroshev conjecture: for both convex and non convex cases, the numerically computed diffusion coefficient D of the actions is compatible with an exponential fit, and the convex case is definitely more stable than the steep one. For four dimensional maps (ν = 3), since we find that in the steep case D(ɛ) has large oscillations around an exponential behaviour, the agreement of our numerical experiments with the conjecture is not sharp, and it is found by considering a sup over different initial conditions.